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In constraint-based causal discovery, the existing algorithms systematically use a series of conditional independence (CI) relations observed in the data to recover an equivalence class of causal graphs in the large sample limit. One limitation of these algorithms is that CI tests lose statistical power as conditioning set size increases with finite samples. Recent research proposes to limit the conditioning set size for robust causal discovery. However, the existing algorithms require exhaustive testing of all CI relations with conditioning set sizes up to a certain integer k. This becomes problematic in practice when variables with large support are present, as it makes CI tests less reliable due to near-deterministic relationships, thereby violating the faithfulness assumption. To address this issue, we propose a causal discovery algorithm that only uses CI tests where the conditioning sets are restricted to a given set of conditioning sets including the empty set C. We call such set of CI relations IC conditionally closed. We define the notion of C-Markov equivalence: two causal graphs are C-Markov equivalent if they entail the same set of CI constraints from IC. We propose a graphical representation of C-Markov equivalence and characterize such equivalence between two causal graphs. Our proposed algorithm called the C-PC algorithm is sound for learning the C-Markov equivalence class. We demonstrate the utility of the proposed algorithm via synthetic and real-world experiments in scenarios where variables with large support or high correlation are present in the data. Our source code is available online at github.com/kenneth-lee-ch/cpc. 

Abstract Groundwater mixing dynamics play a crucial role in the biogeochemical cycling of shallow wetlands. In this paper, we conducted groundwater simulations to investigate the combined effects of evaporation and local heterogeneity on mixing dynamics in shallow wetland sediments. The results show that evaporation causes groundwater and solutes to upwell from deep sediments to the surface. As the solute reaches the surface, evaporation enhances the accumulation of the solute near the surface, resulting in a higher solute concentration than in deep sediments. Mapping of flow topology reveals that local heterogeneity generates spatially varied mixing patterns mainly along preferential flow pathways. The upwelling of groundwater induced by surface evaporation through heterogeneous sediments is likely to create distinct mixing hotspots that differ spatially from those generated by lateral preferential flows driven by large‐scale hydraulic gradients, which enhances the overall mixing in the subsurface. These findings have strong implications for biogeochemical processing in wetlands.